Can a boy know by looking at a girl is going through her period?

No, for the most part a boy cannot tell that a girl is menstruating simply by looking at her.

There can be some ways to tell by looking alone, for example if she is unfortunate enough to experience a leak while the boy is around or her menstrual pads are showing somehow. In some cases girls skin may worsen, such as acne break-outs on the chin, but this isn't true for all girls at all times.